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IJESM Volume 2, Issue 1 ISSN: 2320-0294
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A Quarterly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories
Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-Gage, India as well as in Cabell’s Directories of Publishing Opportunities, U.S.A.
International Journal of Engineering, Science and Mathematics
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2013
Stability Analysis of Mathematical Syn-
Ecological Model Comprising of
Prey-Predator, Host-Commensal, Mutualism
and Neutral Pairs-II
(Three of the Four Species are washed out States)
B. Ravindra Reddy*
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Abstract
This investigation deals with a mathematical model of a four species (S1, S2, S3 and S4)
Syn-Ecological system (Three of the four species are washed out states). S2 is a predator
surviving on the prey S1. The predator S2 is a commensal to the host S3. The pairs S2 and S4, S1
and S3 are neutral. The mathematical model equations characterizing the syn-ecosystem constitute
a set of four first order non-linear coupled differential equations. There are in all sixteen
equilibrium points. Criteria for the asymptotic stability of four of the sixteen equilibrium points:
Three of the four species are washed out states only are established in this paper. The linearized
equations for the perturbations over the equilibrium points are analyzed to establish the criteria
for stability and the trajectories illustrated.
* Department of Mathematics, JNTUH CE, Nachupally, Karimnagar District-505501, India.

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1. INTRODUCTION
Mathematical modeling is an important interdisciplinary activity which involves the
study of some aspects of diverse disciplines. Biology, Epidemiodology, Physiology, Ecology,
Immunology, Bio-economics, Genetics, Pharmocokinetics are some of those disciplines. This
mathematical modeling has raised to the zenith in recent years and spread to all branches of life
and drew the attention of every one. Mathematical modeling of ecosystems was initiated by
Lotka [9] and by Volterra [18]. The general concept of modeling has been presented in the
treatises of Meyer [11], Cushing [4], Paul Colinvaux [11], Freedman [5], Kapur [6, 7]. The
ecological interactions can be broadly classified as Prey-Predation, Competition, Mutualism and
so on. N.C. Srinivas [17] studied the competitive eco-systems of two species and three species
with regard to limited and unlimited resources. Later, Lakshmi Narayan [8] has investigated the
two species prey-predator models. Stability analysis of competitive species was carried out by
Archana Reddy [3] while Acharyulu [1, 2] investigated Ammensalism between two species.
Recently local stability analysis for a two-species ecological mutualism model has been
investigated by present author et al [12, 13, 14, 15, 16]. Example for S1, S2, S3 and S4 are Insects,
Insectivorous Plants (nephantis, drosera etc.), VAM associated with the plant roots, Soil bacteria
respectively.
2. BASIC EQUATIONS
The model equations for a four species multi-system are given by a set of four non-linear ordinary
differential equations as
(i) For S1: The Prey of S1 and Neutral to S3
21
1 1 11 1 12 1 2
dN
a N a N a N N
dt
(2.1)
(ii) For S2: The Predator surviving on S1 and Commensal to S3
22
2 2 22 2 21 2 1 23 2 3
dN
a N a N a N N a N N
dt
(2.2)
(iii) For S3: The Host of S2 and Mutual to S4
23
3 3 33 3 34 3 4
dN
a N a N a N N
dt
(2.3)
(iv) For S4: Mutual to S3 and Neutral to S2
24
4 4 44 4 43 4 3
dN
a N a N a N N
dt
(2.4)
with the following notation.

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Ni (t): Population strengths of the species Si at time t, i=1, 2, 3, 4.
ai : The natural growth rates of Si, i = 1,2,3,4
a12,a21 : Interaction (Prey-Predator) coefficients of S1 due to S2 and S2 due to S1
a13 : Coefficient for commensal for S1 due to the Host S3
a34, a43 : Mutually interaction between S3 and S4
Ki: i
ii
a
a
: Carrying capacities of Si, i=1, 2, 3, 4.
Further the variables N1, N2, N3, N4 are non-negative and the model parameters a1, a2, a3, a4; a11,
a22, a33, a44; a12, a21, a13, a24 are assumed to be non-negative constants.
3. EQUILIBRIUM STATES
The system under investigation has sixteen equilibrium states defined by
0, 1,2,3,4idN
i
dt
(3.1)
are given in the following table.
I. Fully washed out state:
E1: 1 2 3 40, 0, 0, 0N N N N
II. States in which three of the four species are washed out and fourth is surviving
E2: 4
1 2 3 4
44
0, 0, 0,
a
N N N N
a
E3: 3
1 2 3 4
33
0, 0, , 0
a
N N N N
a
E4: 2
1 2 3 4
22
0, , 0, 0
a
N N N N
a
E5: 1
1 2 3 4
11
, 0, 0, 0
a
N N N N
a
III. States in which two of the four species are washed out while the other two are surviving
E6: 4 34 3 44 3 43 4 33
1 2 3 4
33 44 34 43 33 44 34 43
0, 0, ,
a a a a a a a a
N N N N
a a a a a a a a
This state exists only when 33 44 34 43 0a a a a
E7: 2 4
1 2 3 4
22 44
0, , 0,
a a
N N N N
a a
E8: 3 23 2 3
1 2 3 4
22 33 22 33
0, , , 0
a a a a
N N N N
a a a a
E9: 1 4
1 2 3 4
11 44
, 0, 0,
a a
N N N N
a a
E10: 1 3
1 2 3 4
11 33
, 0, , 0
a a
N N N N
a a

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E11: 1 22 2 12 1 21 2 11
1 2 3 4
11 22 12 21 11 22 12 21
, , 0, 0
a a a a a a a a
N N N N
a a a a a a a a
IV. States in which one of the four species is washed out while the other three are surviving
E12:
23 4 34 3 44 2 4 34 3 44
1 2 3
22 33 44 34 43 22 33 44 34 43
4 33 3 43
4
33 44 34 43
( )
0, , ,
( )
a a a a a a a a a a
N N N
a a a a a a a a a a
a a a a
N
a a a a
E13: 1 4 34 3 44 4 33 3 43
1 2 3 4
11 33 44 34 43 33 44 34 43
, 0, ,
a a a a a a a a a
N N N N
a a a a a a a a a
This state exists only when 33 44 34 43( ) 0a a a a
E14: 1 22 2 12 1 21 2 11 4
1 2 3 4
11 22 12 21 11 22 12 21 44
, , 0,
a a a a a a a a a
N N N N
a a a a a a a a a
E15: 4 5 3
1 2 3 4
1 1 33
, , , 0
a
N N N N
a
where
1 33 11 22 12 21 4 33 1 22 2 12 3 23 12
5 33 1 21 2 11 3 23 11
( ), ( )
( )
a a a a a a a a a a a a a
a a a a a a a a
This state exists only when 4 0
V. The co-existent state (or) Normal steady state
E16: 1 12 23 2 4 11 23 2
1 2
3 33 44 34 43 3 33 44 34 43
, ,
( ) ( )
a a a a
N N
a a a a a a a a
4 34 3 44 4 33 3 43
3 4
33 44 34 43 33 44 34 43
,
a a a a a a a a
N N
a a a a a a a a
Where
1 1 22 2 12 33 44 34 43 2 3 44 4 34
3 11 22 12 21 4 1 21 2 11 33 44 34 43
( )( ),
, ( )( )
a a a a a a a a a a a a
a a a a a a a a a a a a
This state exists only when 1 21 2 11 33 44 34 43( ) 0 ( ) 0a a a a and a a a a .
The present paper deals with three of the four species are washed out states only. The stability of
the other equilibrium states will be presented in the forth coming communications.

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4. STABILITY OF THREE OF THE FOUR SPECIES WASHED OUT EQUILIBRIUM
STATES
(Sl. Nos 2,3,4,5 in the above Equilibrium States)
4.1 Stability of the Equilibrium State E2:
4
1 2 3 4
44
0, 0, 0,
a
N N N N
a
(4.1.1)
Substituting (4.1) in (2.1), (2.2), (2.3), (2.4) and neglecting products and higher powers
of 1 2 3 4u , u , u , u , we get
1
1 1
du
a u
dt
(4.1.2) 2
2 2
du
a u
dt
(4.1.3)
3
3 3
du
l u
dt
(4.1.4) 4 43 4
4 4 3
44
du a a
a u u
dt a
(4.1.5)
Here 34 4
3 3
44
a a
l a
a
(4.1.6)
The characteristic equation of which is
1 2 3 4( )( )( )( ) 0a a l a (4.1.7)
The roots a1, a2, 3l are positive and -a4 is negative.
Hence the steady state is unstable.
The solutions of the equations (4.1.2), (4.1.3), (4.1.4), (4.1.5) are
1
1 10
a t
u u e (4.1.8)
2
2 20
a t
u u e (4.1.9)
3
3 30
l t
u u e (4.1.10)
3443 4 30 43 4 30
4 40
44 3 4 44 3 4
[ ]
( ) ( )
l ta ta a u a a u
u u e e
a l a a l a
(4.1.11)
where u10, u20, u30, u40 are the initial values of u1, u2, u3, u4 respectively.
There would arise in all 576 cases depending upon the ordering of the magnitudes of the growth
rates a1, a2, a3, a4 and the initial values of the perturbations u10(t), u20(t), u30(t),u40(t) of the species
S1, S2, S3, S4. Of these 576 situations some typical variations are illustrated through respective
solution curves that would facilitate to make some reasonable observations.
The solutions are illustrated in figures.
Case (i): If u30 < u40 < u20 < u10 and a3 < a2 < a4 < a1
In this case initially S4 dominates the Predator (S2) and the host
(S3) of S2 till the time instant t*
24, t*
34 respectively and
thereafter the dominance is reversed. Also the Predator (S2)
dominates the host (S3) of S2 till the time instant t*
32 and
thereafter the dominance is reversed.
Fig. 1

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Case (ii): If u20 < u30 < u40 < u10 and l3 < a2 < a1 < a4
In this case initially S4 dominates the host (S3) of S2 and the
Predator (S2) till the time instant t*
34, t*
24 respectively and
thereafter the dominance is reversed. Also the host (S3) of S2
dominates the Predator (S2) till the time instant t*
23 and
Fig. 2
Substituting (4.1.1) in (2.1), (2.2), (2.3), (2.4) and neglecting products and higher powers
of 1 2 3 4u , u , u , u , we get
1
1 1
du
a u
dt
(4.2.1) 2
2 2
du
m u
dt
(4.2.2)
3 34 3
3 3 4
33
du a a
a u u
dt a
(4.2.3) 4
4 4
du
n u
dt
(4.2.4)
Here 23 3
2 2
33
a a
m a
a
, 43 3
4 4
33
a a
n a
a
1 2 3 4( )( )( )( ) 0a m a n (4.2.5)
The roots a1, m2, 4n are positive and -a3 is negative.
Hence the steady state is unstable.
The solutions of the equations (4.2.1), (4.2.2), (4.2.3), (4.2.4) are
1
1 10
a t
u u e (4.2.6)
2
2 20
m t
u u e (4.2.7)
3 434 3 40 34 3 40
3 30
33 4 3 33 4 3
[ ]
( ) ( )
a t n ta a u a a u
u u e e
a n a a n a
(4.2.8)
4
4 40
n t
u u e (4.2.9)
Case (i): If u40 < u10 < u20 < u30 and m2 < a3 < a1 < n4
In this case initially the host (S3) of S2 dominates S4 till the time
instant t*
43 and thereafter the dominance is reversed. Also the
Predator (S2) dominates the Prey (S1) and S4 till the time instant
t*
12, t*
42 respectively and the dominance gets reversed thereafter.
Similarly the Prey (S1) dominates S4 till the time instant t*
41 and

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Fig. 3
Case (ii): If u10 < u20 < u30 < u40 and n4 < a1 < a3 < m2
In this case initially S4 dominates the Predator (S2) till the time
instant t*
24 and thereafter the dominance is reversed. Also the
host (S3) of S2 dominates the Predator (S2) and the Prey (S1)
till the time instant t*
23, t*
13 respectively and the
dominance gets reversed thereafter.
Fig. 4
of 1 2 3 4u , u , u , u , we get
1
1 1
du
ru
dt
(4.3.1) 2 21 2 23 2
2 2 1 3
22 22
du a a a a
a u u u
dt a a
(4.3.2)
3
3 3
du
a u
dt
(4.3.3) 4
4 4
du
a u
dt
(4.3.4)
Here 12 2
1 1
22
a a
r a
a
(4.3.5)
1 2 3 4( )( )( )( ) 0r a a a (4.3.6)
Case (A): When 1 0r (i.e., when 1 12
2 22
a a
a a
)
The roots 1r , 2a are negative and 3a , 4a are positive.
Hence the equilibrium state is unstable.
The solutions of the equations (4.3.1) (4.3.2), (4.3.3), (4.3.4) are
1
1 10
rt
u u e (4.3.7)
32 121 2 10 23 2 30 21 2 10 23 2 30
2 20
22 1 2 22 2 3 22 1 2 22 2 3
[ ]
( ) ( ) ( ) ( )
a ta t rta a u a a u a a u a a u
u u e e e
a r a a a a a r a a a a
(4.3.8)
3
3 30
a t
u u e (4.3.9)

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4
4 40
a t
u u e (4.3.10)
Case (i): If u20 < u40 < u10 < u30 and a4 < a3 < a2 < r1
In this case initially the Prey (S1) dominates S4 and the Predator
(S2) till the time instant t*
41, t*
21 respectively and thereafter the
dominance is reversed.
Fig. 5
Case (ii): If u30 < u20 < u40 < u10 and a3 < r1 < a4 < a2
In this case initially the Prey (S1) dominates S4, the Predator
(S2) and the host (S3) of S2 till the time instant t*
41, t*
21, t*
31
respectively and thereafter the dominance is reversed. Also the
Predator (S2) dominates the host (S3) of S2 till the time instant
t*
32 and thereafter the dominance is reversed.
Fig.6
Case (B): When 1 0r (i.e., when 1 12
2 22
a a
a a
)
The roots 1,r 3a , 4a are positive and 2a is negative.
Hence the equilibrium state in unstable.
In this case the solutions are same as in case (A) and the solutions
are illustrated in figures.
Case (i): If u20 < u30 < u40 < u10 and a3 < a4 < a2 < r1
Fig. 7
In this case initially S4 dominates the Predator (S2) till the time instant t*
24 and thereafter the
dominance is reversed. Also the host (S3) of S2 dominates the Predator (S2) till the time instant
t*

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Case (ii): If u40 < u30 < u10 < u20 and a2 < a3 < r1 < a4
In this case initially the Predator (S2) dominates the Prey (S1),
the host (S3) of S2 and S4 till the time instant t*
12, t*
32, t*
42
respectively and thereafter the dominance is reversed. Also the
Prey (S1) dominates S4 till the time instant t*
41 and the
dominance gets reversed thereafter. Similarly the host (S3) of
S2 dominates S4 till the time instant t*
43 and thereafter the
dominance is reversed.
Fig. 8
of 1 2 3 4u , u , u , u , we get
1 12 1
1 1 2
11
du a a
a u u
dt a
(4.4.1) 2
2 2
du
q u
dt
(4.4.2)
3
3 3
du
a u
dt
(4.4.3) 4
4 4
du
a u
dt
(4.4.4)
Here 21 1
2 2
11
a a
q a
a
(4.4.5)
1 2 3 4( )( )( )( ) 0a q a a (4.4.6)
The roots 2 ,q 3a , 4a are positive and 1a is negative.
Hence the equilibrium state in unstable.
The solutions of the equations (4.4.1) (4.4.2), (4.4.3), (4.4.4) are
1 21 12 20 1 12 20
1 10
11 2 1 11 2 1
[ ]
( ) ( )
a t q ta a u a a u
u u e e
a q a a q a
(4.4.7)
2
2 20
q t
u u e (4.4.8) 3
3 30
a t
u u e (4.4.9)
4
4 40
a t
u u e (4.4.10)
Case (i): If u20 < u40 < u30 < u10 and a4 < a1 < q2 < a3
In this case initially the Prey (S1) dominates the host (S3) of S2,
S4 and the Predator (S2) till the time instant t*
31, t*
41, t*
21
respectively and thereafter the dominance is reversed. Also S4

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dominates the Predator (S2) till the time instant t*
Fig. 9
Case (ii): If u30 < u10 < u40 < u20 and a4 < a3 < a1 < q2
In this case initially S4 dominates the host (S3) of S2 till the time
instant t*
34 and the dominance gets reversed thereafter. Also the
Prey (S1) dominates the host (S3) of S2 till the time instant t*
31 and
Fig. 10
REFERENCES
[1] Acharyulu K.V.L.N and Pattabhi Ramacharyulu N. Ch: On The Stability of an Ammensal-
Enemy Harvested Species Pair with Limited Resources, Int. J. Open Problems Compt. Math
(IJOPCM), Vol. 3, No. 2, pp.241-266, June 2010.
[2] Acharyulu K.V.L.N and Rama Gopal N: Numerical Approach to a Mathematical Model of
Three species ecological Ammensalism, Int. J. of Mathematical Archive, 3(6), pp.2273-2282, 2012.
[3] Archana Reddy R: On the stability of some mathematical models in biosciences
interacting species, Ph.D thesis, 2009, JNTU.
[4] Cushing J. M: Integro – differential equations and Delay Models in Population Dynamics,
Lecture Notes in Biomathematics, Vol. 20, Springer – Verlag , Heidelberg, 1977.
[5] Freedman H.I: Deterministic Mathematical Models in Population Ecology, Marcel – Decker,
New York, 1980.
[6] Kapur J.N: Mathematical Modelling, Wiley – Eastern, 1988.
[7] Kapur J.N: Mathematical Models in Biology and Medicine Affiliated East – West, 1985.
[8] Lakshmi Narayan K: A Mathematical study of Prey-Predator Ecological Models with a partial
covers for the prey and alternative food for the predator, Ph.D thesis, 2004, J.N.T.University.
[9] Lotka A.j: Elements of Physical biology, Williams and Wilkins, Baltimore, 1925.

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[10] Meyer W.J: Concepts of Mathematical Modelling, Mc Graw – Hill, 1985.
[11] Paul Colinvaux: Ecology, John Wiley and Sons Inc., New York, 1986.
[12] Ravindra Reddy B.: A Study on mathematical models of ecological mutualism between two
Interacting Species, Ph.D Thesis, O.U, 2008.
[13] Ravindra Reddy B, Lakshmi Narayan K, and Pattabhiramacharyulu N.Ch: A model of two
mutually interacting species with limited resources for both the species. Int. J.of Engg. Research
& Indu. Appls, Vol.2, No.II (2009), pp 281-291.
[14] Ravindra Reddy B, Lakshmi Narayan K, and Pattabhiramacharyulu N.Ch: A model of two
mutually interacting species with limited resources and harvesting of both the species at a
constant rate. International J. of Math. Sci & Engg. Appls. (IJMSEA), Vol. 4, No. III (August,
2010), pp. 97-106.
[15] Ravindra Reddy B, Srilatha R, Lakshmi Narayan K., and Pattabhiramachryulu N.Ch: A
model of two mutually interacting species with Harvesting, Atti Della Fondazione Giorgio
Ronchi, Anno LXVI, 2011 - N. 3, 317-331.
[16] Ravindra Reddy B: A Model of two mutually interacting Species with Mortality Rate for the
Second Species, Advances in Applied Science Research, 2012, 3(2):757-764.
[17] Srinivas N.C: Some Mathematical aspects of modeling in Bio Medical Sciences, Ph.D thesis,
1991, Kakatiya University.
[18] Volterra V: Leconssen la theorie mathematique de la leitte pou lavie, Gauthier – Villars,
Paris, 1931.

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